Cubic graphs induced by bridge trisections
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912034161950720 |
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| author | Meier, Jeffrey Thompson, Abigail Zupan, Alexander |
| author_facet | Meier, Jeffrey Thompson, Abigail Zupan, Alexander |
| contents | Every embedded surface $\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\mathcal{K})$ into three simple pieces. In this case, the surface $\mathcal{K}$ is determined by an embedded 1-complex, called the $\textit{1-skeleton}$ of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph $Γ$ that inherits a natural Tait coloring, a 3-coloring of the edge set of $Γ$ such that each vertex is incident to edges of all three colors. In this paper, we reverse this association: We prove that every Tait-colored cubic graph is isomorphic to the 1-skeleton of a bridge trisection corresponding to an unknotted surface. When the surface is nonorientable, we show that such an embedding exists for every possible normal Euler number. As a corollary, every tri-plane diagram for a knotted surface can be converted to a tri-plane diagram for an unknotted surface via crossing changes and interior Reidemeister moves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2007_07280 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Cubic graphs induced by bridge trisections Meier, Jeffrey Thompson, Abigail Zupan, Alexander Geometric Topology Combinatorics 57Q45, 57M50, 05C15 Every embedded surface $\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\mathcal{K})$ into three simple pieces. In this case, the surface $\mathcal{K}$ is determined by an embedded 1-complex, called the $\textit{1-skeleton}$ of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph $Γ$ that inherits a natural Tait coloring, a 3-coloring of the edge set of $Γ$ such that each vertex is incident to edges of all three colors. In this paper, we reverse this association: We prove that every Tait-colored cubic graph is isomorphic to the 1-skeleton of a bridge trisection corresponding to an unknotted surface. When the surface is nonorientable, we show that such an embedding exists for every possible normal Euler number. As a corollary, every tri-plane diagram for a knotted surface can be converted to a tri-plane diagram for an unknotted surface via crossing changes and interior Reidemeister moves. |
| title | Cubic graphs induced by bridge trisections |
| topic | Geometric Topology Combinatorics 57Q45, 57M50, 05C15 |
| url | https://arxiv.org/abs/2007.07280 |